formes quadratiques

E
Q ϕ
n(n+1)/2
kn, Q(x) = Px2
i
ϕ(x, y)=1/2(q(x+y)−q(x)−q(y)) = 1/4∗(q(x+
y)−q(x−y))
Q
(ϕ(ei, ej)B= (ei)Q
In
P=MB(B0)M0
B(Q) =tP Mb(Q)P
Q(x) = Piaiix2
i+Pi<j aij xixjQ
Q
aii a11 Q(x) = a11x2
1+x1B+C
(x2, . . . , xn) (x2, . . . , xn)Q(x) = a11(x1+B/2a11)2+ [C−B2/4a11]
aii aij
a12 Q(x) = a12x1x2+x1B+x2C+D(x3,...xn)
Q(x) = a12/4∗[(x1+x2+ (B+C)/a12)2−(x1−x2+ (C−B)/a12)2+ [D−(B+C)/a12]

Q B det(MB(Q))
Q
Fq
C
Fqα
q ϕ(x, y)=0
A E
q Et
ϕ(x, .)ϕ(., y)
x7→ ϕ(x, .)y7→ ϕ(., y)
E E∗B, B∗MB(ϕ)
Q MB(Q)
⊥A⊥vectA =⊥A⊥0 = E⊥(A∪B) =⊥A∩⊥B⊥A+⊥B⊂⊥
(A∩B)dim(H) + dim(⊥H) = dimE +dim(f∩Ker Q)
⊥⊥A=E dim(A)+dim(⊥A) =
dimE ⊥A
q(x) = 0
Ker(q)⊂C(q)
F F F ⊥F F ⊥
E
⇔F⊂F⊥2dim(F)< E

(P, q)q(e1)=0, q(e2)=0 f(e1, e2)6= 0
(P, q)
q
E
Pr
q
(ei, εi)
e1,...er, ε1,...εr0Ir
Ir0
Ir0
0−Ir
H
H
F F 0E
u∈O(q)u(F) = F0
q|Fq|F0
σ q|Fq|F0
q v(q)
E H G H
G2v(q) = dim(H)
O(q)qa
q
q O(q)u q(u(x) = q(x)O+(q)
det(u) = 1
F g gug−1
g(F)
O(q)
n>3So(q)

E n Q
Q(x) = f1(x)+. . .+fp(x)−fp+1(x)−. . .−fp+q(x)f
Q Q(x) = f0
1(x)+. . .+f0
p0(x)−f0 p0+1(x)−. . .−f0 p0+q0(x)
p=p0q=q0
Q(p, q)
p+q=rgQ
E F +, F −F0Q−
E Q F +F−
(dimF +, dimF −)
Q=λ1L2
1+. . . +λpL2
p−λp+1L2
p+1 −. . . −λp +qL2
p+q
(p, q)
Q(x, y, z, t) = xy +yz +xz +yz = 1/4∗((x+y+z+t)2+ (x+z−y−t)2)
(1,1) 2
Q(x, y, z) = x2−2y2+xz +yz = (x+z/2)2−2(y−z/4)2−z2/8
3
B E A =MB(Q)A
diag(λi)
(Card {λi>0}, Card {λi<0})
Q Q0
B Q−MB(Q0)
E Q
B T r(MB(ϕ)=0
C(Q) = Ker(Q)
Ker(Q)⊂C(Q)
Q min(p, q)
(p, p)

y2= 2px 0Ox (0, p/2)
x2/a2+y2/b2= 1, a 6b Ox
b e =c/a c =p(a2−b2)
p=b2/a (0, c) (0,−c)
x2/a2−y2/b2= 0 e=c/a c =p(a2+b2)
p=b2/a (0, c) (0,−c)
2
Q
(x−a)2+−(y−b)2=c
13x2+ 64y2+ 62xy = 1
x2+y2= 0 x2+y2=−1x2−y2= 0
x2/a2+y2/b2+z2/c2= 1
x2/a2+y2/b2−z2/c2= 0
x2/a2+y2/b2−z2/c2= 1
x2/a2+y2/b2−z2/c2=−1
x2/a2+y2/b2= 1
x2/a2−y2/b2= 1
x2= 2pz
x2/a2+y2/b2=z
x2/a2−y2/b2=z
2
4xy −z2−1=0
4xy −z2−1+2x−4y= 0
E f
U E
Lc(E, E0)'(E×E, R)
d2f(h, k) =< Hf(x)h, k >